A Free Boundary Problem in an Unbounded Domain and Subsonic Jet Flows from Divergent Nozzles

SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6337-6360, October 2024.
Abstract. This paper concerns subsonic jet flows from two-dimensional finitely long divergent nozzles with straight solid walls, which are governed by a free boundary problem for a quasilinear elliptic equation. It is assumed that the angle of the nozzle and the location of the inlet are fixed, while the length of the nozzle is free. For a given surrounding pressure and a given incoming mass flux, it is shown that there is a critical number not greater than [math] for the angle of the nozzle such that there exists a unique subsonic jet flow if the angle of the nozzle is less than the critical number. If this critical number is less than [math], then there is not a subsonic jet flow when the angle of the nozzle takes this critical number; furthermore, as the angle of the nozzle tends to this critical number, either the length of the nozzle tends to zero, or a sonic point will occur at the inlet. Moreover, it is shown that the subsonic jet flow tends to a uniform horizontal flow exponentially at the downstream. As to the jet, it is smooth away from the connecting point with the wall of the nozzle, and it connects the wall of the nozzle with [math] regularity for each exponent [math]. Furthermore, the jet is strictly concave to the fluid and tends to a line parallel to the symmetrical axis exponentially.